Finite Group Databasebeta

← #11724468

Subgroups — #11724468

Order 864.

442 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
4#64264—312nomaximal
5#64151—31nomaximal
6#6312—41nomaximal
7C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C32×Q8D311yescharacteristic, complemented
9#3828—31no—
10C2×C3×SL(2,𝔽3)D3112yescomplemented
11C3⋊GL(2,𝔽3)—312no—
12C3⋊#226—312no—
13#3756—31no—
14#3756—31no—
15#3755—31no—
16#3755—31no—
17#3748—31no—
18C22×C33—41no—
19C2×C3⋊(C3⋊D3)—41no—
20C3⋊(C3⋊Dic3)—41no—
21#983—99no—
22#942—94no—
23C32×Q8—31no—
24C32×Q8C2×D311yescharacteristic
25C2×C32×C4—31no—
26#638—61no—
27#638—1212no—
28C4⋊(C3⋊D3)—31no—
29C4×C3⋊D3—61no—
30#634—31no—
31C3×SL(2,𝔽3)C2×D3112yes—
32C3⋊(C3⋊C8)—31no—
33C3⋊(C3⋊C8)—31no—
34C2×C33—41no—
35C2×C33—81no—
36C3⋊(C3⋊D3)—81no—
37C2×C3×Q8C3⋊D314yescomplemented
38#235—94no—
39C2×SL(2,𝔽3)C3⋊D319yescomplemented
40GL(2,𝔽3)—99no—
41#226—99no—
42#216—94no—
43#216—94no—
44#215—94no—
45#215—94no—
46#208—94no—
47C22×C32—412no—
48C22×C32S411yescharacteristic
49C2×C3⋊D3—61no—
50C2×C3⋊D3—1212no—
51C32×C4—31no—
52C32×C4—31no—
53C3⋊Dic3—61no—
54C3⋊Dic3—1212no—
55#137—271noSylow
56C33—41noSylow
57C3×Q8—34no—
58C3×Q8C2×C3⋊D314yes—
59C2×C3×C4—34no—
60#67—184no—
61#67—369no—
62D12—94no—
63C4×D3—184no—
64Dic6—94no—
65SL(2,𝔽3)C2×C3⋊D319yes—
66C3⋊C8—94no—
67C3⋊C8—94no—
68C2×C32—21no—
69C2×C32—412no—
70C2×C32—812no—
71C2×C32C2×S411yescharacteristic
72C3⋊D3—121no—
73C3⋊D3—2412no—
74#41—271no—
75C2×Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
76Q16—271no—
77Q16—271no—
78SD16—271no—
79SD16—271no—
80M16—271no—
81C22×C3—49no—
82C22×C3C3⋊S414yes—
83C2×D3—184no—
84C2×D3—369no—
85C3×C4—34no—
86C3×C4—34no—
87Dic3—184no—
88Dic3—369no—
89C32—412no—
90C32#98311yescharacteristic, complemented
91Q8—31no—
92Q8—271no—
93Q8C2×C3⋊(C3⋊D3)11yescharacteristic
94D4—271no—
95D4—541no—
96C2×C4—31no—
97C2×C4—541no—
98C8—271no—
99C8—271no—
100C2×C3—24no—
101C2×C3—49no—
102C2×C3—89no—
103C2×C3C2×C3⋊S414yes—
104D3—364no—
105D3—729no—
106C22—541no—
107C22C3⋊(C3⋊S4)11yescharacteristic
108C4—31no—
109C4—31no—
110C4—541no—
111C3—49no—
112C3#6426414yescomplemented
113C2—21no—
114C2—1081no—
115C2C2×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C22
FittingC2×C32×Q8D3
socleC2×C32C2×S4
hypercenterC22C3⋊(C3⋊S4)
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C32×SL(2,𝔽3)C2semidirect
2. C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C3⋊(C3⋊#226)C2semidirect
8. C2×C32×Q8D3semidirect
10. C2×C3×SL(2,𝔽3)D3semidirect
37. C2×C3×Q8C3⋊D3semidirect
39. C2×SL(2,𝔽3)C3⋊D3semidirect
75. C2×Q8C3⋊(C3⋊D3)semidirect
90. C32#983semidirect
112. C3#64264semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C32×SL(2,𝔽3)2
C3⋊(C3⋊GL(2,𝔽3))2
C3⋊(C3⋊#226)2
#642643
#641513
#63124